The 2D manipulator domain studied here directly extends to 3D manipulator scenarios for both static and mobile manipulators. In the case of static manipulators, the end-effector is approximated as a sphere. Each link is approximated as a set of spheres kept side by side. However, in heavily cluttered environments such an approximation can be computationally intensive since each sphere has to be checked for collision with obstacles. An alternative and effective approach is to consider the minimum-volume enclosing ellipsoid for each link [1]}. It is known that for every convex polyhedron, there exits a unique ellipsoid of minimal volume that contains the polyhedron and is called the Löwner-John ellipsoid of the polyhedron [2]}. Thus each link can be represented by their corresponding Löwner-John ellipsoids. The distance between two ellipsoids is used to modify the collision condition in (REF ). For mobile manipulators, the collision condition should also checked for the base as discussed in the 2D robot section.
